This paper will primarily be concerned with the rank 2 semi-simple split Lie group \(G=\mathrm {PSL}(2,\mathbb {R}) \times \mathrm {PSL}(2,\mathbb {R})\) . The product structure of \(G\)  makes our study more amenable towards techniques from classical Teichmüller theory. For \(S\)  a closed, orientable, smooth surface of genus \(g>1\) , work of Goldman [1]} shows the connected components of the character variety \(\chi (\pi _{1}(S), \mathrm {PSL}(2, \mathbb {R}))\)  are determined by the Euler number. In particular, the distinguished component with maximal Euler number of \(2g-2\)  is the Teichmüller space Teich\((S)\) . If we denote the character variety for \(G= \mathrm {PSL}(2,\mathbb {R})\times \mathrm {PSL}(2,\mathbb {R})\)  by \(\chi (\pi _{1}(S), G)\) , then the connected components are merely products of the connected components of \(\chi (\pi _{1}(S), \mathrm {PSL}(2, \mathbb {R}))\) . The maximal component Max\((S,G)\)  of \(\chi (\pi _{1}(S), G)\)  is collection of conjugacy classes of pairs of representations, each of which is a Fuchsian representation. Hence \(\mathrm {Max}(S):=\mathrm {Max}(S, \mathrm {PSL}(2,\mathbb {R})\times \mathrm {PSL}(2,\mathbb {R}))\)  is the product of two copies of Teichmüller space.
